Abstract:
Certain spaces of X-valued sequences are introduced and some of their properties are investigated. K the- Toeplitz duals of these spaces are examined.

Abstract:
The ± and 2-duals spaces of generalized ￠ “p spaces are characterized, where 0 Keywords generalized K the-Toeplitz dual spaces --- sequences of linear operators --- generalized p spaces.

Abstract:
In this paper we define the sequence spaces S ￠ “ ￠ (p), Sc(p) and Sc0(p) and determine the K the-Toeplitz duals of S ￠ “ ￠ (p). We also obtain necessary and sufficient conditions for a matrix A to map S ￠ “ ￠ (p) to ￠ “ ￠ and investigate some related problems.

Abstract:
Let $S$ be the shift operator on the Hardy space $H^2$ and let $S^*$ be its adjoint. A closed subspace $\FF$ of $H^2$ is said to be nearly $S^*$-invariant if every element $f\in\FF$ with $f(0)=0$ satisfies $S^*f\in\FF$. In particular, the kernels of Toeplitz operators are nearly $S^*$-invariant subspaces. Hitt gave the description of these subspaces. They are of the form $\FF=g (H^2\ominus u H^2)$ with $g\in H^2$ and $u$ inner, $u(0)=0$. A very particular fact is that the operator of multiplication by $g$ acts as an isometry on $H^2\ominus uH^2$. Sarason obtained a characterization of the functions $g$ which act isometrically on $H^2\ominus uH^2$. Hayashi obtained the link between the symbol $\phii$ of a Toeplitz operator and the functions $g$ and $u$ to ensure that a given subspace $\FF=gK_u$ is the kernel of $T_\phii$. Chalendar, Chevrot and Partington studied the nearly $S^*$-invariant subspaces for vector-valued functions. In this paper, we investigate the generalization of Sarason's and Hayashi's results in the vector-valued context.

Abstract:
We give a necessary and a sufficient condition for the boundedness of the Toeplitz product $T_FT_{G^*}$ on the vector valued Bergman space $L_a^2(\mathbb{C}^n)$, where $F$ and $G$ are matrix symbols with scalar valued Bergman space entries. The results generalize existing results in the scalar valued Bergman space case. We also characterize boundedness and invertibility of Toeplitz products $T_FT_{G^*}$ in terms of the Berezin transform, generalizing results found by Zheng and Stroethoff for the scalar valued Bergman space.

Abstract:
The β-dual of a vector-valued sequence space is defined and studied. We show that if an X-valued sequence space E is a BK-space having AK property, then the dual space of E and its β-dual are isometrically isomorphic. We also give characterizations of β-dual of vector-valued sequence spaces of Maddox ℓ(X,p), ℓ∞(X,p), c0(X,p), and c(X,p).

Abstract:
let μ be a normal scalar sequence space which is a k-space under the family of semi-norms m and let x be a locally convex space whose topology is generated by the family of semi-norms x. the space μ{x} is the space of all x valued sequences c = {ck} such that {q(ck)} ？μ{x} for all q ？ x. the space μ{x} is given the locally convex topology generated by the semi-norms eppq(c) = p({q(ck)}), p ？ x, q ？ m. we show that if μ satisfies a certain multiplier type of gliding hump property, then pointwise bounded subsets of the a-dual of μ{x} with respect to a locally convex space are uniformly bounded on bounded subsets of μ{x}

Abstract:
Let μ be a normal scalar sequence space which is a K-space under the family of semi-norms M and let X be a locally convex space whose topology is generated by the family of semi-norms X. The space μ{X} is the space of all X valued sequences chi = {c k} such that {q(c k)} μ{X} for all q X. The space μ{X} is given the locally convex topology generated by the semi-norms eppq(chi) = p({q(c k)}), p X, q M. We show that if μ satisfies a certain multiplier type of gliding hump property, then pointwise bounded subsets of the a-dual of μ{X} with respect to a locally convex space are uniformly bounded on bounded subsets of μ{X}

Abstract:
We introduce generalized Lorentz difference sequence spaces . Also we study some topologic properties of this space and obtain some inclusion relations. 1. Introduction Throughout this work, , , and denote the set of positive integers, real numbers, and complex numbers, respectively. The notion of difference sequence space was introduced by K？zmaz in [1] in 1981 as follows: for , , where for all . Et and ？olak in [2] defined the sequence space for , , where , , for all , and showed that this space is a Banach space with norm Subsequently difference sequence spaces has been discussed in Ahmad and Mursaleen [3], Malkowsky and Parashar [4], Et and Basarir [5], and others. Let be a Banach space. The Lorentz sequence space for , is the collection of all sequences such that is finite, where is nonincreasing rearrangement of (we can interpret that the decreasing rearrangement is obtained by rearranging in decreasing order). This space was introduced by Miyazaki in [6] and examined comprehensively by Kato in [7]. A weight sequence is a positive decreasing sequence such that , and , where for every . Popa [8] defined the generalized Lorentz sequence space for as follows: where ranges over all permutations of the positive integers and is a weight sequence. It is known that and hence for each there exists a nonincreasing rearrangement of and (see [8, 9]). Let be a Banach space and let be a weight sequence. We introduce the vector-valued generalized Lorentz difference sequence space for . The space is the collection of all -valued -sequences ？？ such that is finite, where is nonincreasing rearrangement of and for all . We will need the following lemmas. Lemma 1 (see [10]). Let and be the nonincreasing and nondecreasing rearrangements of a finite sequence of positive numbers, respectively. Then for two sequences and of positive numbers we have Lemma 2 (see [7]). Let be an -valued double sequence such that for each and let be an -valued sequence such that (uniformly in ). Then and for each where and are the nonincreasing rearrangements of and , respectively. 2. Main Results Theorem 3. The space for is a linear space over the field or . Proof. Let and let and be the nonincreasing rearrangements of the sequences and , respectively. Since is nonincreasing, by Lemma 1 we have where . Let . Hence we get This shows that and so is a linear space. Theorem 4. The space for is normed space with the norm where denotes the nonincreasing rearrangements of . Proof. It is clear that . Let . Then we have and for all . Hence we get . Let . Since weight sequence is decreasing, by Lemma

Abstract:
We introduce the vector-valued sequence spaces , , and , and , using a sequence of modulus functions and the multiplier sequence of nonzero complex numbers. We give some relations related to these sequence spaces. It is also shown that if a sequence is strongly -Cesàro summable with respect to the modulus function then it is -statistically convergent.